Monthly Archives: September 2007

Two Tracts by Galois

While updating my post on Hilbert, thanks to Manfred Karbe’s e-mails I came across these two very interesting, powerful and engaging pieces on abstraction and instruction in mathematics by Galois. I put these texts here just in case the French website where they are taken from becomes unavailable sometime. I think the ultimate reference for Galois’ writings is the book `Écrits et Mémoires Mathématiques d’Évariste Galois’ (Gauthier-Villars, Paris 1962). It contains all of Galois’ writings including his work on abelian integrals. The second tract appears in pages 21-25 in this book. The first tract, only partly reproduced here, appears in full in pages 3-11 of this book.

Préface

“ Cecy est un livre de bonne foy. Montagne.

…………………….Si avec aussi peu de chances d’être compris, je publie, malgré tout, le fruit de mes veilles, c’est afin de prendre date pour mes recherches, c’est afin que les amis que j’ai formés dans le monde avant qu’on m’enterrât sous les verrous, sachent que je suis bien en vie, c’est peut-être aussi dans l’espérance que ces recherches pourront tomber entre les mains de personnes à qui une morgue stupide n’en interdira pas la lecture, et les diriger dans la nouvelle voie que doit, selon moi, suivre l’analyse dans ses branches les plus hautes. Il faut bien savoir que je ne parle ici que d’analyse pure; mes assertions transportées aux applications les plus directes des mathématiques deviendraient paradoxales.

Les longs calculs algébriques ont d’abord été peu nécessaires au progrès des Mathématiques, les théorèmes fort simples gagnaient à peine à être traduits dans la langue de l’analyse. Ce n’est guère que depuis Euler que cette langue plus brève est devenue indispensable à la nouvelle extension que ce grand géomètre a donnée à la science. Depuis Euler les calculs sont devenus de plus en plus nécessaires, mais de plus en plus difficiles à mesure qu’ils s’appliquaient à des objets de science plus avancés. Dès le commencement de ce siècle, l’algorithme avait atteint un degré de complication tel que tout progrès était devenu impossible par ce moyen, sans l’élégance que les géomètres modernes ont su imprimer à leurs recherches, et au moyen de laquelle l’esprit saisit promptement et d’un seul coup un grand nombre d’opérations.

Il est évident que l’élégance si vantée et à si juste titre, n’a pas d’autre but.

Du fait bien constaté que les efforts des géomètres les plus avancés ont pour objet l’élégance, on peut donc conclure avec certitude qu’il devient de plus en plus nécessaire d’embrasser plusieurs opérations à la fois, parce que l’esprit n’a plus le temps de s’arrêter aux détails.

Or je crois que les simplifications produites par l’élégance des calculs, (simplifications intellectuelles, s’entend; de matérielles il n’y en a pas) ont leurs limites; je crois que le moment arrivera où les transformations algébriques prévues par les spéculations des analystes ne trouveront plus ni le temps ni la place de se produire; à tel point qu’il faudra se contenter de les avoir prévues. Je ne veux pas dire qu’il n’y a plus rien de nouveau pour l’analyse sans ce secours: mais je crois qu’un jour sans cela tout serait épuisé.

Sauter à pieds joints sur ces calculs; grouper les opérations, les classer suivant leurs difficultés et non suivant leurs formes; telle est, suivant moi, la mission des géomètres futurs; telle est la voie où je suis entré dans cet ouvrage.

Il ne faut pas confondre l’opinion que j’émets ici, avec l’affectation que certaines personnes ont d’éviter en apparence toute espèce de calcul, en traduisant par des phrases fort longues ce qui s’exprime très brièvement par l’algèbre, et ajoutant ainsi á la longueur des opérations, les longueurs d’un langage qui n’est pas fait pour les exprimer. Ces personnes-là sont en arrière de cent ans.

Ici rien de semblable; ici on fait l’analyse de l’analyse: ici les calculs les plus élevés exécutés jusqu’à présent sont considérés comme des cas particuliers, qu’il a été utile, indispensable de traiter, mais qu’il serait funeste de ne pas abandonner pour des recherches plus larges. Il sera temps d’effectuer des calculs prévus par cette haute analyse et classés suivant leurs difficultés, mais non spécifiés dans leur forme, quand la spécialité d’une question les réclamera.

La thèse générale que j’avance ne pourra être bien comprise que quand on lira attentivement mon ouvrage qui en est une application: non que ce point de vue théorique ait précédé l’application; mais je me suis demandé, mon livre terminé, ce qui le rendrait si étrange à la plupart des lecteurs, et rentrant en moi-même, j’ai cru observer cette tendance de mon esprit à éviter les calculs dans les sujets que je traitais, et qui plus est, j’ai reconnu une difficulté insurmontable à qui voudrait les effectuer généralement dans les matières que j’ai traitées.

On doit prévoir que, traitant des sujets aussi nouveaux, hasardé dans une voie aussi insolite, bien souvent des difficultés se sont présentées que je n’ai pu vaincre. Aussi dans ces deux mémoires et surtout dans le second qui est plus récent, trouvera-t-on souvent la formule “je ne sais pas”). La classe des lecteurs dont j’ai parlé au commencement ne manquera pas d’y trouver à rire. C’est que malheureusement on ne se doute pas que le livre le plus précieux du plus savant serait celui où il dirait tout ce qu’il ne sait pas, c’est qu’on ne se doute pas qu’un auteur ne nuit jamais tant à ses lecteurs que quand il dissimule une difficulté. Quand la concurrence c’est-à-dire l’égoïsme ne règnera plus dans les sciences, quand on s’associera pour étudier, au lieu d’envoyer aux académies des paquets cachetés, on s’empressera de publier ses moindres observations pour peu qu’elles soient nouvelles, et on ajoutera: “je ne sais pas le reste”.

De Ste Pélagie, décembre 1831,

EVARISTE GALOIS. “

The second tract by Galois:

“Sur l’Enseignement des Sciences, des Professeurs, des Ouvrages, des Examinateurs

MONSIEUR LE RÉDACTEUR,
Je vous serais obligé, si vous voulez bien accueillir les réflexions suivantes, relatives à l’étude des mathématiques dans les collèges de Paris.

D’abord dans les sciences, les opinions ne comptent pour rien; les places ne sauraient être la récompense de telle ou telle manière de voir en politique ou en religion. Je m’informe si un professeur est bon ou mauvais, et je m’inquiète fort peu de sa façon de penser dans des matières étrangères à ses études scientifiques. Ce n’était donc pas sans douleur et indignation que, sous le gouvernement de la restauration, on voyait les places devenir la proie des plus offrants en fait d’idées monarchiques et religieuses. Cet état de choses n’est pas changé; la médiocrité, qui fait preuve de sa répugnance pour le nouvel ordre de choses, est encore privilégiée; et cependant les opinions ne devraient pas être mises en ligne de compte, lorsqu’il s’agit d’apprécier le mérite scientifique des individus.

Commençons par les collèges; là les élèves de mathématiques se destinent pour la plupart à l’école polytechnique; que fait-on pour les mettre en état d’atteindre ce but ? Cherche-t-on à leur faire concevoir le véritable esprit de la science par l’exposé des méthodes les plus simples ? Fait-on en sorte que le raisonnement devienne pour eux une seconde mémoire ? N’y aura-t-il pas au contraire quelque ressemblance entre la manière dont ils APPRENNENT les mathématiques et la manière dont ils APPRENNENT les leçons de français et de latin ? Jadis un élève aurait appris d’un professeur tout ce qui lui est utile de savoir; maintenant il faut le supplément de un, de deux répétiteurs pour préparer un candidat à l’école polytechnique.

Jusques à quand les pauvres jeunes gens seront-ils obligés d’écouter ou de répéter toute la journée ? Quand leur laissera-t-on du temps pour méditer sur cet amas de connaissances, pour coordonner cette foule de propositions sans suite, de calculs sans liaison ? N’y aurait-il pas quelque avantage à exiger des élèves les mêmes méthodes, les mêmes calculs, les mêmes formes de raisonnement, s’ils étaient à la fois les plus simples et les plus féconds ? Mais non, on enseigne minutieusement des théories tronquées et chargées de réflexions inutiles, tandis qu’on omet les propositions les plus simples et les plus brillantes de l’algèbre; au lieu de cela, on démontre à grands frais de calculs et de raisonnements toujours longs, quelquefois faux, des corollaires dont la démonstration se fait d’elle-même.

D’où vient le mal ? Assurément ce n’est pas des professeurs des collèges; ils montrent tous un zèle fort louable; ils sont les premiers à gémir de ce qu’on ait fait de l’enseignement des mathématiques un véritable métier. La cause du mal, c’est aux libraires de MM. les examinateurs qu’il faut la demander. Les libraires veulent de gros volumes: plus il y a de choses dans les ouvrages des examinateurs, plus ils sont certains d’une vente fructueuse; voilà pourquoi nous voyons apparaître chaque année ces volumineuses compilations où l’on voit les travaux défigurés des grands maîtres à côté des essais de l’écolier.

D’un autre côté, pourquoi les examinateurs ne posent-ils les questions aux candidats que d’une manière entortillée ? Il semblerait qu’ils craignissent d’être compris de ceux qu’ils interrogent; d’où vient cette malheureuse habitude de compliquer les questions de difficultés artificielles ? Croit-on donc la science trop facile ? Aussi qu’arrive-t-il ? L’élève est moins occupé de s’instruire que de passer son examen. Il lui faut sur chaque théorie une RÉPÉTITION de chacun des quatre examinateurs; il doit apprendre les methodes qu’ils affectionnent, et savoir à l’avance, pour chaque question et chaque examinateur, quelles doivent être ses réponses et même son maintien. Aussi il est vrai de dire qu’on a fondé depuis quelques années une science nouvelle qui va grandissant chaque jour, et qui consiste dans la connaissance des dégoûts et des préférences scientifiques, des manies et de l’humeur de MM. les examinateurs.

Etes-vous assez heureux pour sortir vainqueur de l’épreuve ? Etes-vous enfin désigné comme l’un des deux cents géomètres à qui on porte les armes dans Paris ? Vous croyez être au bout: vous vous trompez, c’est ce que je vous ferai voir dans une prochaine lettre.

E. G. “

——————————————————————————–

Notes:
Cette lettre d’Evariste Galois, fut publiée dans Gazette des Ecoles: Journal de l’Instruction Publique, de l’Université, des Séminaires, numéro 110, 2e année, Janvier 1831.
Aucune deuxième lettre n’a été publiée.

Les motifs – ou le coeur dans le coeur

It is with this fascinating title that A. Grothendieck presents in Recoltes et Semailles (cfr. Promenade à travers une oeuvre ou l’Enfant et la Mère) the subject of motives: the deepest of the twelve research themes around which he developed his “long-run” research program that literally revolutionized the field of algebraic geometry in the decade 1958-68. Motives were envisaged as the “heart of the heart” of the new geometry (arithmetic geometry) that Grothendieck invented following a scientific strategy based on the introduction of a series of new concepts organized on a progressive level of generality: starting with schemes, topos and sites then continuing with the yoga of motives and motivic Galois groups and finally introducing anabelian algebraic geometry and Galois-Teichmuller theory.
If the notions of scheme and topos were the two crucial ideas which constituted the original driving force in the development of this new geometry — Grothendieck was evidently fascinated by the concepts of geometric point, space and symmetry — it is only with the notion of a motive that one eventually captures the deepest structure, the heart of the profound identity between geometry and arithmetic.

Grothendienck wrote very little about motives. The foundations are documented in his unpublished manuscript “Motifs” and were discussed on a seminar at the Institut des Hautes Études Scientifiques, in 1967. We know, by reading Recoltes et Semailles, that he started thinking about motives in 1963-64. J.P. Serre has included in his paper “Motifs” an extract from a letter that Grothendieck wrote to him in August 1964 in which he talks (rather vaguely, in fact) of the notions of motive, fiber functor, motivic Galois group and weights.
Motives were introduced with the ultimate goal to supply an intrinsic explanation for the analogies occurring among the various cohomological theories for algebraic varieties: they were expected to play the role of a universal cohomological theory (the motivic cohomology) and also to furnish a linearization of the theory of algebraic varieties, by eventually providing (this was Grothendieck’s viewpoint) the correct framework for a successful attack to the Weil’s Conjectures on the zeta-function of an algebraic variety over a finite field.
Unlike in the framework of algebraic topology where the standard cohomological functor is uniquely characterized by the Eilenberg-Steenrod axioms in terms of the normalization associated to the value of the functor on a point, in algebraic geometry there is no suitable cohomological theory with integers coefficients, for varieties defined over a field k, unless one provides an embedding of k into the complex numbers. In fact, by means of such mapping one can form the topological space of the complex points of the original algebraic variety and finally compute the Betti (singular) cohomology. This construction however, does in general depend upon the choice of the embedding of k in the field of complex numbers. Moreover, Hodge cohomology, algebraic de-Rham cohomology, étale l-adic cohomology furnish several examples of different cohomology functors which can be simultaneously associated to a given algebraic variety, each of which supplying a relevant information on the topological space.
Grothendieck theorized that this plethora of different cohomological data should be somewhat encoded systematically within a unique and more general theory of cohomological nature that acts as an internal “liaison” between algebraic geometry and the collection of available cohomological theories. This is the idea of the “motif”, namely the common reason behind this multitude of cohomological invariants which governs and controls systematically all the cohomological apparatus pertaining to an algebraic variety or more in general to ascheme.
The original construction of a category M of (pure) motives over a field k starts with two preliminary considerations. The first consideration is that M should be the target of a natural contravariant functor connecting the category C of smooth, projective algebraic varieties over a field k to M. Such functor should map an object X in C to its associated motive M(X). The second consideration is that this functor should, by construction, factor through any particular cohomological theory.
Now, keeping in mind this goal, one thinks about the axiomatizing process of a cohomological theory in algebraic geometry. This is done by introducing a contravariant functor X -> H(X) from C to a graded abelian category, where the sets of morphisms between its objects form K-vector spaces (K is a field of characteristic zero, that for simplicity, I fix here equal to the rationals). One also would like that any correspondence V–> W (an algebraic cycle in the cartesian product VxW that can be view as the graph of a multi-valued algebraic mapping) induces contravariantly, a mapping on cohomology and that the target category is suitably defined so that it contains among its objects any “Weil cohomological theory”, namely a cohomology which satisfies among other axioms Poincaré duality and Künneth formula.
This preliminary disquisition helps one in formalizing the construction of the category of motives by following a three-steps procedure. One wishes to enlarges the category C in a precise way with the hope to produce also an abelian category. The three steps are shortly resumed as follows.
(1) One moves from C to a category with the same objects but where the sets of morphisms are the equivalence classes of rational correspondences. Here, the natural choice of the equivalence relation is the numerical equivalence relation as it is the coarsest one among the possible relations between algebraic cycles which can be seen to induce, via the cohomological axioms of any Weil cohomological theory, well-defined homomorphisms in cohomology.
(2) One enlarges the collection of objects of the category defined in (1), by formally adding kernels and images of projectors. This step is technically referred to as the “pseudo-abelian envelope” of the category defined in (1) and it is motivated by the expectation to define an abelian category of motives in which for instance, the Künneth formula can be applied.
(3) Finally, one considers the opposite of the category defined in (2).
Now, after having diligently applied all this abstract machinery, one would like to see a fruitful application of these ideas, in the form, for instance, of the proof of a major conjecture. However, one also perceives quite soon that a successful application of the yoga of motives is subordinated to a thorough knowledge of the theory of algebraic cycles, since the construction of the category M is centered on the idea of enlarging the sets of morphisms by implementing the notion of correspondence. It is for this reason that the Standard Conjectures (cohomological criteria for the existence of interesting algebraic cycles) were associated, since the beginning, to the theory of motives as they seem to play the “conditio sine qua non” a theory of motives has a concrete and successful application.
However, in order to put the Standard Conjectures in the right perspective and to avoid perhaps, an over-estimation of their importance, one should also record that Y. Manin gave in 1968, the first interesting application of these ideas on motives by producing an elegant proof of the Riemann-Weil hypothesis for non-singular three-dimensional projective unirational varieties over a finite field, without appealing to the Standard Conjectures. Moreover, we also know that the Weil’s Conjectures have been proved by P. Deligne in 1974 without using neither the theory of motives nor the Standard Conjectures.
Almost forty years have passed since these ideas were informally discussed in the “Grothendieck’s circle”. An enlarged and in part still conjectural theory of mixed motives has in the meanwhile proved its usefulness in explaining conceptually, some intriguing phenomena arising in several areas of pure mathematics, such as Hodge theory, K-theory, algebraic cycles, polylogarithms, L-functions, Galois representations etc.
Very recently, some new applications of the theory of motives to number-theory and quantum field theory have been found or are about to be developed, with the support of techniques supplied by noncommutative geometry and the theory of operator algebras.
In number-theory, a conceptual understanding of the interpretation proposed by A. Connes of the Weil explicit formulae as a Lefschetz trace formula over the noncommutative space of adèle classes, requires the introduction of a generalized category of motives which is inclusive of spaces which are highly singular from a classical viewpoint. Several questions arise already when one considers special types of zero-dimensional noncommutative spaces, such as the space underlying the quantum statistical dynamical system defined by J.B. Bost and Connes in their paper “Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking” (Selecta Math. (3) 1995). This space is a simplified version of the adèle classes and it encodes in its group of symmetries, the arithmetic of the maximal abelian extension of the rationals.
A new theory of endomotives (algebraic and analytic) has been recently developed in “Noncommutative geometry and motives: the thermodynamics of endomotives” (to appear in Advances in Mathematics). The objects of the category of endomotives are noncommutative spaces described by semigroup actions on projective limits of Artin motives (these are among the easiest examples of pure motives, as they are associated to zero-dimensional algebraic varieties). The morphisms in this new category generalize the notion of (algebraic) correspondences and are defined by means of étale groupoids to account for the presence of the semigroup actions.
An open and interesting problem is connected to the definition of a higher dimensional theory of noncommutative motives and in particular the set-up of a theory of noncommutative elliptic motives and modular forms.
A suitable generalization of the yoga of motives to noncommutative geometry has already produced some interesting results in the form, for example, of an analog in characteristic zero of the action of the Weil group on the étale cohomology of an algebraic variety.
It seems quite exciting to pursue these ideas further: the hope is that the motivic techniques, once suitably transferred in the framework of noncommutative geometry may supply useful tools and produce even more substantial applications than those obtained in the classical commutative context.
What is the “heart of the heart” of noncommutative geometry?

New books on noncommutative geometry

This seems to be a banner year for books on noncommutative geometry. Leading the pack is “Noncommutative Geometry, Quantum Fields. and Motives” by Alain and Matilde which will be published by the American Mathematical Society in January 2008. Earlier this year we introduced two other books.

Now two other new books that just appeared in the market. Topological and Bivariant K-Theory by Joachim Cuntz, Ralf Meyer, and Jonathan Rosenberg has just been published by Birkhaeuser. In publisher’s introduction we read:
“Topological K-theory is one of the most important invariants for noncommutative algebras equipped with a suitable topology or bornology. Bott periodicity, homotopy invariance, and various long exact sequences distinguish it from algebraic K-theory.
We describe a bivariant K-theory for bornological algebras, which provides a vast generalization of topological K-theory. In addition, we discuss other approaches to bivariant K-theories for operator algebras. As applications, we study K-theory of crossed products, the Baum-Connes assembly map, twisted K-theory with some of its applications, and some variants of the Atiyah-Singer Index Theorem. “

The second book is Local and Analytic Cyclic Homology by Ralf Meyer and is published by the European Mathematical Society Publication House. Here is an excerpt by the publisher describing the book: “Periodic cyclic homology is a homology theory for non-commutative algebras that plays a similar role in non-commutative geometry as de Rham cohomology for smooth manifolds. While it produces good results for algebras of smooth or polynomial functions, it fails for bigger algebras such as most Banach algebras or C*-algebras. Analytic and local cyclic homology are variants of periodic cyclic homology that work better for such algebras. In this book the author develops and compares these theories, emphasising their homological properties. This includes the excision theorem, invariance under passage to certain dense subalgebras, a Universal Coefficient Theorem that relates them to K-theory, and the Chern–Connes character for K-theory and K-homology”

A note to our readers and authors: if you know of any other books on the subject that is published this year, or will be published soon, please let one of us know. I know of at least one, but we should wait a bit!

Wir müssen wissen, wir werden wissen!

With the above words (translation: we must know, we will know), Hilbert ended his 1930 address in Königsberg, at the Congress of the Association of German Natural Scientists and Medical Doctors. A four minute excerpt was broadcast by radio, and is available here. This I believe is one of the earliest audio files of a speech by a mathematician of note, available online.
For the original German text and an English translation click here. The above picture was kindly sent by A. Rivero. (See his interesting comments about the place the picture was taken). Many thanks to him!

Now that we are inaugurating a new section in the blog under the label `multimedia’ (see Matilde’s recent post for an example), I thought time is right to add something I always wanted to share with the readers of this blog. Hilbert’s address is not about NCG of course! Some of the relevant underlying philosophical and cultural aspects of the Zeitgeist challenged by Hilbert in his address are briefly discussed here. Notice, however, that Hilbert can be regarded as one of the great grandfathers of NCG and the subject owes a lot to him and his Göttingen school of functional analysis and spectral theory in the years 1900-1912 (Erhard Schmidt, Hermann Weyl, ….; as well as Otto Toeplitz who was not in Göttingen). Hilbert’s work was centered around the theory of integral equations and its allied spectral theory as it was mostly motivated by Fredholm’s 1900 papers. An immediate dramatic success was Weyl’s assymptotic law for the eigenvalues of the Dirichlet problem for Laplacian on bounded domains. This basic result of spectral geometry is one of the foundation stones of NCG as well.

Later abstractions and the development of functional analysis on Hilbert space, by von Neumann and others, led to the theory of operator algebras which is of course one of the sources of NCG.

It is interesting to note that Hilbert’s address was just one year before Gödel’s incompleteness theorems , which in a way showed that, at least globally, one can not be totally optimistic about the power of formalistic approach in mathematics!

Update (Sept 20, 2007): Manfred Karbe kindly wrote in to share the following interesting information (my sincere thanks to him):

`After reading your post and hearing Hilbert’s voice (in the
unmistakable Low Prussian dialect) I luckily found the “Hilbert
Gedenkband” on one of my book shelves, edited by Reidemeister and
published by Springer in 1971……….Attached to this booklet (which is out of print now, no wonder at those times!) was a record which contains the four
minute excerpt that was broadcast by radio. See also
this public appeal to locate the original recording’

You Tube QFT

For the readers who may be interested in the work Alain and I did on renormalization and the Riemann-Hilbert correspondence, as well as in a general introduction to the Connes-Kreimer theory, I noticed recently that the series of lectures of the minicourse we gave two years ago at Vanderbilt University have now been put on You Tube (or Google Video).

You can find them here:

Lecture 1 (1 h 52 min) by Matilde

Lecture 2 (1 h 7 min) by Alain

Lecture 3 (1 h 10 min) by Alain

Lecture 4 (1 h) by Matilde

Lecture 5 (1 h 30 min) by Matilde

Lecture 6 (1 h 25 min) by Alain

Lecture 7 (1 h 24 min) by Alain

Lecture 8 (1 h 7 min) by Matilde

News on K-front

Today the editorial board of the new Journal of K-theory put out a public statement, which we reproduce below:

STATEMENT OF THE EDITORS OF THE ” JOURNAL OF K-THEORY »

After several public statements and news articles regarding the Springer journal “K-theory” (KT), and the new “Journal of K-Theory” (JKT) to be distributed by Cambridge University Press (CUP), the mathematical community has become aware of ongoing changes. On behalf of the entire Editorial Board of the new JKT, we want to give as precise a picture of the situation as we can at the moment, especially to the authors.

It is very important to us that the authors should not suffer as a result of the transition.
Those authors who submitted papers to KT before August 2007, regardless of whether the paper has already been accepted or is just awaiting review, have three choices:
1) Choose another journal.
2) Maintain submission with KT for final review if necessary and publication if accepted.
3) Transfer their article to the new JKT.
All authors who have not yet done so should please notify Professor Bak on the one hand, Professors Lueck and Ranicki on the other hand, about their choice, as soon as possible. For those who opt for choice #2, Professors Lueck and Ranicki have promised to take over the remaining editorial duties.
We can guarantee that the authors who choose option (3) will have a smooth transition, with their articles progressing as if there has been no change. We will also do everything we can to help those who choose options (1) and (2). In particular, if the authors instruct us, we will be happy to forward to the journals of their choice the full information regarding the status of their articles.

In 2004, because of growing dissatisfaction with Springer, the editorial board of KT authorized Prof. Anthony Bak, the Editor in Chief, to begin negotiations with other publishers. The editorial board was unhappy with the poor quality of the work done by Springer, for example the huge number of misprints in the published version of the articles, the long delay in publication and the high prices Springer was charging.
The negotiations came to a conclusion in 2007. A new journal, entitled “Journal of K-theory” (JKT) will commence publication in late 2007. It will be printed by Cambridge University Press. Papers will appear earlier online, as ‘forthcoming articles’.

The title of JKT is currently owned by a private company. This situation is only meant as a temporary solution to restart publication of K-theory articles as soon as possible. It is the Board’s intention to create a non-profit academic foundation and to transfer ownership of JKT to this foundation, as soon as possible, but no later than by the end of 2009, a delay justified by many practical considerations.

This shift towards more academic control of journals is not new. We follow here a path opened by Compositio Mathematica, Commentarii Mathematici Helvetici, and others (see for instance the interesting paper of Gerard van der Geer which appeared in the Notices of the AMS in May 2004). We believe that such changes can help keep prices low.

We trust in Prof. Bak’s leadership for the launching of JKT and forming, together with the editorial board, the foundation to house the Journal. The statutes of the foundation will provide democratic rules governing the future course and development of the journal, including the election of the managing team.

We hope to have provided a fair picture of the current situation, and we plan to issue another public statement when new developments come up. In case of further questions, please contact any of the signatories.

Let us conclude from a broader perspective: The editorial board is committed to secure the journal’s quality and long-term sustainability.

Signatures
A. Bak
P. Balmer
S. Bloch
G. Carlsson
A. Connes
E. Friedlander
M. Hopkins
B. Kahn
M. Karoubi
G. Kasparov
A. Merkurjev
A. Neeman
T. Porter
J. Rosenberg
A. Suslin
Guoping Tang
B. Totaro
V. Voevodsky
C. Weibel
Guoliang Yu